LDR 02179 2200253 4500 010 _a9783540668091 011 _a1617-9692 090 _a14032 101 _aeng 100 _a20181212 frey50 200 _aElemental methods in ergodic Ramsey theory _bNUM _fRandall McCutcheon 210 _aBerlin _cSpringer _d1999 225 _9180169 _aLecture notes in mathematics _i(Online) _x1617-9692 _v1722 300 _aThis book gives an overview over the main results of chromatic and density Ramsey theory in ℤ. It is based on lecture notes for a graduate course and therefore additionally provides a large variety of exercises and problems that are helpful for a better understanding of the field. Even though many of the presented results were originally proven combinatorially, most of the proofs in this book are based on topological dynamics and ergodic theory. The problems considered mainly fall into the classes of van der Waerden type theorems, Hindman type theorems and Szemerédi type theorems. After an introduction to Ramsey theory and to topological dynamics in the first chapter, results from infinitary Ramsey theory and density Ramsey theory are presented in Chapters 2 and 3. In Chapters 4 and 5 ergodic theory is used to establish double recurrence results and multiple recurrence results. In particular, Roth theorems and Szeméredi theorems are proven in this context. (Zentralblatt) 686 _20 _9161347 _a05D10 _bCombinatorics -- Extremal combinatorics _xRamsey theory 686 _20 _9165058 _a54H20 _bGeneral topology -- Connections with other structures, applications _xTopological dynamics 686 _20 _9162897 _a28D05 _bMeasure and integration -- Measure-theoretic ergodic theory _xMeasure-preserving transformations 686 _20 _9163817 _a37A05 _bDynamical systems and ergodic theory -- Ergodic theory _xMeasure-preserving transformations 700 _4070 _aMcCutcheon _bRandall _f1965- _9180347 856 _uhttp://link.springer.com/book/10.1007/BFb0093959 _zSpringerlink 856 _uhttp://zbmath.org/?q=an:0945.05063 _zZentralblatt 856 _uhttp://www.ams.org/mathscinet-getitem?mr=1738544 _zMathSciNet 905 _aaw _b2014-01-06 001 14032